QUANTUM MEASUREMENT THEORY AND ITS APPLICATIONS

Transcription

1 QUANTUM MEASUREMENT THEORY AND ITS APPLICATIONS Recent experimental advances in the control of quantum superconducting circuits, nanomechanical resonators and photonic crystals have meant that quantum measurement theory is now an indispensable part of the modeling and design of experimental technologies. This book, aimed at graduate students and researchers in physics, gives a thorough introduction to the basic theory of quantum measurement and many of its important modern applications. Measurement and control is explicitly treated in superconducting circuits and optical and optomechanical systems, and methods for deriving the Hamiltonians of superconducting circuits are introduced in detail. Further applications covered include feedback control, metrology, open systems and thermal environments, Maxwell s demon, and the quantum-to-classical transition. KURT JACOBS is an Associate Professor of Physics at the University of Massachusetts at Boston. He is a leading researcher in quantum measurement theory and feedback control, and applications in nano-electromechanical systems. He is author of the textbook Stochastic Processes for Physicists: Understanding Noisy Systems (Cambridge University Press, 2010).

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3 QUANTUM MEASUREMENT THEORY AND ITS APPLICATIONS KURT JACOBS University of Massachusetts at Boston

4 University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. Information on this title: / c 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Jacobs, Kurt (Kurt Aaron), author. Quantum measurement theory and its applications /, University of Massachusetts at Boston. pages cm Includes bibliographical references and index. ISBN (hardback) 1. Quantum measure theory. I. Title. QC M4J dc ISBN Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

5 To my mother, Sandra Jacobs, for many things. Not least for the Nelson Southern Link Decision, a great triumph unsung.

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7 Contents Preface page xi 1 Quantum measurement theory Introduction and overview Classical measurement theory Understanding Bayes theorem Multiple measurements and Gaussian distributions Prior states-of-knowledge and invariance Quantum measurement theory The measurement postulate Quantum states-of-knowledge: density matrices Quantum measurements Understanding quantum measurements Relationship to classical measurements Measurements of observables and resolving power A measurement of position The polar decomposition: bare measurements and feedback Describing measurements within unitary evolution Inefficient measurements Measurements on ensembles of states 40 2 Useful concepts from information theory Quantifying information The entropy The mutual information Quantifying uncertainty about a quantum system The von Neumann entropy Majorization and density matrices Ensembles corresponding to a density matrix Quantum measurements and information Information-theoretic properties Quantifying disturbance 72 vii

8 viii Contents 2.4 Distinguishing quantum states Fidelity of quantum operations 82 3 Continuous measurement Continuous measurements with Gaussian noise Classical continuous measurements Gaussian quantum continuous measurements When the SME is the classical Kalman Bucy filter The power spectrum of the measurement record Solving for the evolution: the linear form of the SME The dynamics of measurement: diffusion gradients Quantum jumps Distinguishing quantum from classical Continuous measurements on ensembles of systems Measurements that count events: detecting photons Homodyning: from counting to Gaussian noise Continuous measurements with more exotic noise? The Heisenberg picture: inputs, outputs, and spectra Heisenberg-picture techniques for linear systems Equations of motion for Gaussian states Calculating the power spectrum of the measurement record Parameter estimation: the hybrid master equation An example: distinguishing two quantum states Statistical mechanics, open systems, and measurement Statistical mechanics Thermodynamic entropy and the Boltzmann distribution Entropy and information: Landauer s erasure principle Thermodynamics with measurements: Maxwell s demon Thermalization I: the origin of irreversibility A new insight: the Boltzmann distribution from typicality Hamiltonian typicality Thermalization II: useful models Weak damping: the Redfield master equation Redfield equation for time-dependent or interacting systems Baths and continuous measurements Wavefunction Monte Carlo simulation methods Strong damping: master equations and beyond The quantum-to-classical transition Irreversibility and the quantum measurement problem Quantum feedback control Introduction Measurements versus coherent interactions Explicit implementations of continuous-time feedback 239

9 Contents ix Feedback via continuous measurements Coherent feedback via unitary interactions Coherent feedback via one-way fields Mixing one-way fields with unitary interactions: a coherent version of Markovian feedback Feedback control via continuous measurements Rapid purification protocols Control via measurement back-action Near-optimal feedback control for a single qubit? Summary Optimization Bellman s equation and the HJB equation Optimal control for linear quantum systems Optimal control for nonlinear quantum systems Metrology Metrology of single quantities The Cramér Rao bound Optimizing the Cramér Rao bound Resources and limits to precision Adaptive measurements Metrology of signals Quantum-mechanics-free subsystems Oscillator-mediated force detection Quantum mesoscopic systems I: circuits and measurements Superconducting circuits Procedure for obtaining the circuit Lagrangian (short method) Resonance and the rotating-wave approximation Superconducting harmonic oscillators Superconducting nonlinear oscillators and qubits The Josephson junction The Cooper-pair box and the transmon Coupling qubits to resonators The RF-SQUID and flux qubits Electromechanical systems Optomechanical systems Measuring mesoscopic systems Amplifiers and continuous measurements Translating between experiment and theory Implementing a continuous measurement Quantum transducers and nonlinear measurements Quantum mesoscopic systems II: measurement and control Open-loop control 383

10 x Contents Fast state-swapping for oscillators Preparing non-classical states Measurement-based feedback control Cooling using linear feedback control Squeezing using linear feedback control Coherent feedback control The resolved-sideband cooling method Resolved-sideband cooling via one-way fields Optimal cooling and state-preparation 416 Appendix A The tensor product and partial trace 432 Appendix B A fast-track introduction for experimentalists 441 Appendix C A quick introduction to Ito calculus 448 Appendix D Operators for qubits and modes 451 Appendix E Dictionary of measurements 456 Appendix F Input output theory 458 F.1 A mode of an optical or electrical cavity 458 F.2 The traveling-wave fields at x = 0: the input and output signals 462 F.3 The Heisenberg equations of motion for the system 463 F.4 A weakly damped oscillator 467 F.5 Sign conventions for input output theory 467 F.6 The quantum noise equations for the system: Ito calculus 468 F.7 Obtaining the Redfield master equation 469 F.8 Spectrum of the measurement signal 470 Appendix G Various formulae and techniques 475 G.1 The relationship between Hz and s 1, and writing decay rates in Hz 475 G.2 Position representation of a pure Gaussian state 475 G.3 The multivariate Gaussian distribution 476 G.4 The rotating-wave approximation (RWA) 476 G.5 Suppression of off-resonant transitions 477 G.6 Recursion relations for time-independent perturbation theory 478 G.7 Finding operator transformation, reordering, and splitting relations 479 G.8 The Haar measure 484 G.9 General form of the Kushner Stratonovich equation 485 G.10 Obtaining steady states for linear open systems 486 Appendix H Some proofs and derivations 490 H.1 The Schumacher Westmoreland Wootters theorem 490 H.2 The operator-sum representation for quantum evolution 492 H.3 Derivation of the Wiseman Milburn Markovian feedback SME 494 References 498 Index 539

11 Preface I would like to thank here a number of people to whom I am indebted in one way or another. To begin, there are five people from whose insights I especially benefited in my formative years in physics. In order of appearance: Sze M. Tan, for teaching me classical measurement theory, and introducing me to information theory and thermodynamics; Howard M. Wiseman, for teaching me about quantum measurement theory; SalmanHabib, for teaching me about open systems and classical chaos; Tanmoy Bhattacharya, for enlightenment on a great variety of topics, and especially for the insight that measurement is driven by diffusion gradients; Gerard Jungman, for mathematical and physical insights, and for introducing me to many beautiful curiosities. I am very grateful to a number of people who helped directly to make this book what it is: Os Vy, Luciano Silvestri, Benjamin Cruikshank, Alexandre Zagoskin, Gelo Tabia, Justin Finn, Josh Combes, Tauno Palomaki, Andreas Nunnenkamp, and Sai Vinjanampathy who read various chapters and provided valuable suggestions that improved the book. Xiaoting Wang who derived Eqs. (G.43) (G.48). Jason Ralph who enlightened me on some superconductor facts that were strangely difficult to extract from the literature. Jason also helped me with the brief history of superconductivity and quantum superconducting circuits in Chapter 7. Justin Guttermuth who saved our asses when we had a house to move into, rooms to paint, a new baby, and I had this book to finish. My colleagues in the UMass Boston physics department for their support especially Maxim Olchanyi, Bala Sundaram, Vanja Dunjco, and Steve Arnason. And last but not least, my wonderful wife Jacqueline, who helped me with the figures and the cover, and put up with the long hours this book required. I apologize in advance for any errors and inadvertent omissions in this book. I would be most grateful to be notified of any errors that you may find. I will include corrections to all errors, as they are found, in an errata file on my website. I am very grateful to a number of readers who sent me corrections for my previous book, Stochastic Processes for Physicists, all of which have been made in the current printing. I was also able to acknowledge these readers in the current printing, which due to their efforts now appears to be largely error-free. While I have endeavored to cite a fairly comprehensive and representative set of research papers on the topics I have covered in this text, it is likely that I have omitted some that xi

12 xii Preface deserve to be included. If you discover that your important paper on topic X has been missed, please send me the reference and I will be glad to correct this omission in any further edition. Finally, it is a pleasure to acknowledge an ARO MURI grant, W911NF , that was led by Daniel Lidar and administered by Harry Chang. This grant provided partial support for a number of research projects during the writing of this book, and from which it greatly benefited.